Axiom ax-mulass 7975

Description: Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axmulass 7933. Proofs should normally use mulass 8003 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Assertion

Ref	Expression
ax-mulass	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

Detailed syntax breakdown of Axiom ax-mulass

Step	Hyp	Ref	Expression
1		cA	. . . 4 class 𝐴
2		cc 7870	. . . 4 class ℂ
3	1, 2	wcel 2164	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2164	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2164	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 980	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		cmul 7877	. . . . 5 class ·
10	1, 4, 9	co 5918	. . . 4 class (𝐴 · 𝐵)
11	10, 6, 9	co 5918	. . 3 class ((𝐴 · 𝐵) · 𝐶)
12	4, 6, 9	co 5918	. . . 4 class (𝐵 · 𝐶)
13	1, 12, 9	co 5918	. . 3 class (𝐴 · (𝐵 · 𝐶))
14	11, 13	wceq 1364	. 2 wff ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))
15	8, 14	wi 4	1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

This axiom is referenced by:  mulass  8003